Tandem Lens Optical Setup

• Tandem lens optical setups are configurations where multiple sequential lens elements collaboratively enhance imaging performance, aberration correction, and beam shaping.
• They employ matrix formalism and both discrete and continuous optimization strategies to compute effective focal lengths and fine-tune inter-element spacing.
• Applications include compact imaging, laser-plasma interactions, multifocal spectral separation, and integrated dynamic focus control for advanced optical systems.

A tandem lens optical setup refers to an arrangement in which two or more optical elements (typically lenses, which may be refractive, diffractive, or metasurface-based) are placed in sequence along an optical axis to achieve properties or functionalities unattainable by single-element systems. The interconnected use of multiple lenses enables precise control of imaging parameters such as focal length, field of view (FOV), aberration correction, beam shaping, multifocal performance, and spectral or spatial selectivity. Tandem lens systems appear throughout modern optics, including imaging, light shaping, advanced communication, and laser–plasma interaction.

1. Foundational Principles and Geometric Configuration

Tandem lens systems function by leveraging the collective action of multiple lens elements to realize composite optical transfer or wavefront properties not possible with a single component. These systems often exploit:

• The cumulative effect of lens power, where the effective focal length is derived from the individual powers and separations of the component lenses.
• The spatial mapping of phase and/or intensity, as in diffractive and metasurface configurations, to customize focus, field shape, or chromatic dispersive properties.
• Principles of energy conservation and ray mapping to redistribute input beam spatial profiles or intensities with high precision, as demonstrated in high-power laser shaping for plasma generation.

A general tandem system employs either monolithic integration (nano- or microfabricated stacked lenses), cascaded discrete elements (commercial optics), or multi-functional flat-optical elements in serial configuration.

2. Matrix Formalism and Effective Imaging Properties

The paraxial analysis of tandem (multi-lens) systems is comprehensively addressed using optical matrix (ABCD) formalism, applicable to both thin and thick lenses. For an arbitrary cascade of $N$ thick lenses, the total optical transfer matrix $M_{\rm total}$ is the product of the individual lens and propagation (translation) matrices:

$$M_{\rm total} = M_N \cdot T_{N-1,N} \cdot \ldots \cdot M_1$$

where each $M_j$ and $T_{j-1,j}$ encapsulate the specific refraction and translation (thickness, air gaps) properties.

When object and image distances are redefined relative to the principal planes of the composite system (which are calculated from the elements of $M_{\rm total}$), the otherwise complex multi-lens configuration reduces to a standard Gaussian imaging equation:

$$\frac{1}{s_o'} + \frac{1}{s_i'} = \frac{1}{f_{\rm eff}}$$

with $f_{\rm eff}$ extracted from $(M_{\rm total})_{12} = -1/f_{\rm eff}$, generalizing to any number of cascaded thick lenses (Callegari, 2021). This result is critical for the analysis and design of practical tandem systems, revealing that the system can often be modeled as a single effective lens (with associated principal planes), greatly simplifying optical modeling without loss of generality.

3. Automated and Discrete Optimization of Tandem Configurations

High-performance tandem lens systems can be developed via automated computational design. A representative example is the "Lens Factory" system (Sun et al., 2015), which operates as follows:

1. Initialization: User specifies high-level constraints (e.g., focal length, FOV, f-number, sensor size). The system selects a canonical starting design (such as a triplet or Double Gauss) from a catalog of off-the-shelf elements, with each component restricted to a power/diameter window around baseline values.
2. Discrete Optimization: The algorithm explores the combinatorial space via discrete element substitution and systematic "splitting" of a lens element into two, subject to:
   • Power conservation in sign and approximate distribution: for element with power $p_0$ split into $p_1$, $p_2$,

   $$(1 - \alpha)\frac{|p_0|}{2} < |p_i| < (1 + \alpha)\frac{|p_0|}{2},\quad i=1,2$$

   with typical $\alpha=0.25$. - Diameter constraint within ±25% of the original. - Curvature reduction heuristics.

This evolution grows the system from simple to more complex tandem configurations, improving aberration correction and imaging performance. Discrete optimization is augmented by intelligent pruning based on FOV and focus compatibility.

1. Continuous Optimization: Fine tuning of inter-element spacings and sensor position using objective functions, typically based on spot size or modulation transfer function (MTF),

$$\mathcal{F}(\mathcal{L}^k_{\mathbf{c},\mathbf{d}, d_k}) = \frac{1}{3n} \sum_{j=1}^3 \sum_{i=1}^{n} f\big(e_i, \boldsymbol{\lambda}_j, \mathcal{L}^k_{\mathbf{c},\mathbf{d}, d_k}\big)$$

where $\mathcal{L}^k_{\mathbf{c},\mathbf{d}, d_k}$ is the $k$-element tandem lens configuration. Techniques such as gradient descent yield assemblies that maximize image sharpness and maintain specified FOV and focal length.

Case studies demonstrate pronounced MTF improvement: a micro 4/3 lens evolved from a 3-element triplet to a 6-element tandem design nearly doubled MTF50 performance (Sun et al., 2015).

4. Advanced Functionality: Beam Manipulation, Spectral, and Multifocal Capabilities

Tandem lens setups extend beyond standard imaging:

• Beam Shaping and Plasmas: In ultrafast laser applications for plasma wakefield acceleration, tandem diffractive optics are designed to reshape intensity and phase to achieve meter-long Bessel foci with tailored on-axis profiles, directly controlling the longitudinal plasma density ramps critical for electron beam matching (Ariniello et al., 1 Sep 2025). This is accomplished by precise phase computation for both lenses:
  • Conservation mapping: $$2\pi I_{\rm in}(r_a) r_a dr_a = 2\pi I_{\rm out}(r_b) r_b dr_b$$
  • Phase encoding:

  $$\phi_A(r) = k_0 \int_0^r dr_a \frac{r_b(r_a) - r_a}{\sqrt{L^2 + [r_b(r_a) - r_a]^2}}$$

  for the first lens, and

  $\phi_B(r) = \phi_r(r) - \phi_{\rm in}(r)$

  for the second, where $\phi_r(r)$ is the required output phase.

• Multi-beam/Double-path Analyzers: Tandem lens–mirror systems can spatially split a focal plane into dual optical paths (direct and indirect), each with independently tunable magnification and filtering (Popowicz et al., 2016). Unlike beam splitters, this preserves photon budget and enables dual-focus, dual-wavelength, or high-dynamic-range imaging.
• Dual-band and Multifocal Operation: Flat diffractive tandem lenses, with radially programmed phase profiles in concentric zones, can realize simultaneous focusing of distinct spectral bands (e.g., visible, NIR) at separate focal planes (Britton et al., 2020). This property enables spectral separation of depth information in imaging, with measured focusing efficiencies of 24% (visible) and 15% (NIR), and polarization insensitivity due to geometric symmetry.
• Metasurface and Zoom Systems: Integrating multi-element tandem architectures at the metasurface or microfabrication level yields ultracompact, dual-FOV metalenses with spin-dependent zooming properties (Zheng et al., 2016), or continuously zoomable telescopes realized via rotationally tunable pairs of toroidal (saddle) lenses, where magnification is strictly a function of component rotation, not lens translation (Bernet, 2018). For saddle-lens pairs, the effective power is governed by

$f_C^{-1} = 2 f_s^{-1} \cos(2\alpha),$

with $\alpha$ the mutual rotation angle.

5. Optical Efficiency and Layered System Optimization

In complex multilayered tandem setups (e.g., perovskite/Si tandem solar cells), per-layer optical losses are substantial. Implementing the transfer matrix (TM) method captures absorption, reflection, and interference more accurately than the Beer–Lambert law, enabling optimization of optical interfaces (such as the inclusion of conductive ITO layers at recombination junctions) for enhanced throughput and reduced loss (Rafieipour et al., 2023). TM analysis yields realistic efficiency estimates (e.g., 23.10% with ITO compared to an overestimated 19.81% without), and can be generalized to complex tandem lens arrangements experiencing multilayer reflection/scattering effects.

6. Monolithic and Adaptive Tandem Systems

Recent advances in 3D nano-printing allow fabrication of monolithic tandem optical systems integrating mechanical actuation, electromagnetic components, and optical elements into a unified structure (Lux et al., 17 Jan 2025). This approach offers:

• Inherent alignment of optical and mechanical axes, essential for maintaining system performance.
• Dynamic focus control through integrated actuators (e.g., lens and magnet mounted on orthoplanar springs actuated with micro-coils), where focus position $\Delta z$ responds to applied magnetic force and spring constant as $\Delta z = F_{z,m}/k_{\rm eff}$.
• High mechanical robustness (resonant frequencies >300 Hz) and micron-scale displacement fidelity.
• Integration potential in tandem lens architectures for real-time focus or aberration compensation.

Table: Core Implementation Aspects Across Tandem Lens Setups

Application Domain	Key Tandem Principle	Performance / Outcome
Imaging (general)	Cascade of standard/diffractive lenses	Aberration control, FOV/focal tuning
Beam shaping (plasma)	Diffractive lens pair with energy/phase mapping	Meter-scale tailored Bessel foci, on-axis intensity shaping
Dual-band imaging	Multi-zone diffractive lens	Simultaneous spatial/spectral separation
Massive MIMO comms	Lens for beam domain channel separation	Asymptotic channel orthogonality, rate scaling
Monolithic actuators	Integrated mechanical/optical design	Compact dynamic focusing, inherent alignment

7. Advances, Applications, and Prospects

Tandem lens systems are central to numerous applications: compact cameras, high-dynamic-range and multi-spectral image acquisition, laser beam shaping for advanced accelerators, optical communication links exploiting spatial multiplexing, and adaptive/integrated photonic devices. Rapid advances in automated design optimization, flat and metasurface optics, and monolithic integration are pushing performance, flexibility, and manufacturability. While the paraxial matrix model provides a universal reduction for cascaded lens systems, practical implementations often require detailed phase engineering, loss modeling, and optimization across discrete and continuous design spaces.

Ongoing directions include expanding broadband and polarization-insensitive operation in flat/tandem diffractive systems, scaling up device apertures, integrating active feedback and focus adjustment, and furthering multilayer transmission/reflection engineering for loss mitigation. The tandem lens paradigm continues to unlock new regimes in optical system function, size, and adaptability.